〜てもいい

て-form + もいい means "it's all right to": 帰ってもいいです, "you may go home". As a question it asks permission: 座ってもいいですか, "may I sit here?"

Literally "even if (you) do it, it's good", which is why the も is there. In speech it's often shortened to ていい: 座っていい?

Common answers: はい、どうぞ ("yes, go ahead"), or to refuse politely, すみません、ちょっと… or ここはだめなんです.

The opposite, "you mustn't", is てはいけない, the next point. And "you don't have to" is なくてもいい, which is built the same way on the ない form.